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Tao: Open math problems being non-renewably mined by AI

mathstodon.xyz

101–110 of 340 posts

Re: Tao: Open math problems being non-renewably mined by AI

#101
> We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential.

In other words, the "right" people need to solve it: the mathematicians who made it their job and not the people working to push AI models forward?

Struggling to understand how a solution to a millennium problem like this isn't a net positive. Presumably Open AI employs mathematicians in these efforts anyway. And I can think of far worse uses of the AI compute resources.

Re: Tao: Open math problems being non-renewably mined by AI

#102
post #52

I'm with @nilesh on this one, and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge. If a problem is "solved" (say, symbolically verified) without any insights gained, it doesn't seem very interesting to the profession. Navier-Stokes is a bit different (because there's a prize attached, so "scooping" matters), but almost all interesting problems don't have any prizes…

Humanity is very biased for the culmination of work, considering everything that comes before and after busywork for the lower masses. Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication? If we move the goal from "find the solution" to "clear up the LLMs work" that doesn't bode well neither for the attractiveness of the problem nor for the career o…

>Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?

I don’t think this is true, especially for novel or unexpected results. I suppose it depends on what you mean by scientifically, and there is a debate in the philosophy of science about what the value of research even is, but a successful replication does not result in substantial updates to one’s beliefs in the way new research does. And if the goal of science is to change our beliefs and bring them closer to what is “real”, successful replications can’t be as valuable as the initial research almost by definition.

Re: Tao: Open math problems being non-renewably mined by AI

#103
post #75

I didn't realize that open math problems were a finite resource. I recall a story about some famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest. Clearly Tao knows a hell of a lot more than I do about this, but I'm surprised that math that close to completion.

> I didn't realize that open math problems were a finite resource.

There's an interesting commentary about this: https://mathstodon.xyz/@tao/117237320796901560

> famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest.

Web search turns up Gauss's comment, with a bit more nuance: "I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of." (https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/quot...)

Re: Tao: Open math problems being non-renewably mined by AI

#104
post #76
post #64

Earlier quoted context omitted.

> Mathematicians will be less likely to work on a problem if there is a solution Yes, that is Tao's premise, I'm just not sure I buy it. Suppose an oracle existed which could answer any question truthfully. Let's ignore the mechanics of this for now, but it could say things like "the Riemann hypothesis is False" or whatever and we would take it as gospel. Does this mean that we wouldn't have mathematicians or physici…

I think his point is that AI is not creating new problems. It may solve "the Riemann hypothesis" but may completely fail to posit a "Mythos hypothesis" which is vital to advance the field. In fact, achieving the former may make the latter even harder because it will disincentivize production of human mathematics which has till now been the only source of "interesting" problems. FWIW this is my understanding of his ar…

Have we asked AI to create new interesting math problems? XD

Re: Tao: Open math problems being non-renewably mined by AI

#105

> We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential. In other words, the "right" people need to solve it: the mathematicians who made it their job and not the people working to push AI models forward? Struggling to understand how a solution to a millennium proble…

If there are no incentives for mathematicians to work on and share progress in tough problems because they will get scooped, then they will end up quitting and in net we will see less progress overall.

Re: Tao: Open math problems being non-renewably mined by AI

#106
post #90
post #75

I didn't realize that open math problems were a finite resource. I recall a story about some famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest. Clearly Tao knows a hell of a lot more than I do about this, but I'm surprised that math that close to completion.

They aren't, but the problem is that open problems tend to emerge when people are working on other problems. If fewer people are spending time deeply thinking about current problems since a handful of labs are solving them with AI without an eye towards understanding and only on verification, the pool of open problems won't be continuously growing. There is a fear that there will be a chilling effect on the community…

So we can let the ai generate some math problems based on the solutions found? Other fields (computer science, physics, ...) can generate math problems too.

Re: Tao: Open math problems being non-renewably mined by AI

#107
post #75

I didn't realize that open math problems were a finite resource. I recall a story about some famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest. Clearly Tao knows a hell of a lot more than I do about this, but I'm surprised that math that close to completion.

I think you can't have read the thread. The whole point is that there is no end of mathematics, an infinite sea; but the constitution of an 'open math problem' is a delicate piece of mathematical thought, at any moment a small supply of drinking water developed by finitely many human being.

Re: Tao: Open math problems being non-renewably mined by AI

#108

Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.

Pure math is practiced mostly for the intellectual thrills and recognition among a very small group of peers. There's little else to it. You don't become rich, you don't become a celebrity. You teach students, write papers, and probably know most other people who work in the same subfield as you. Tao is a sort of a celebrity of the quarter on HN, but I promise you that outside this forum, almost no one has ever heard of him.

If you take that away and turn math into a less fulfilling pursuit where you mostly try to make sense of the output of an LLM, and it's "Astra's theorem #18398" and not "John Doe's last theorem", I'd wager that far fewer people will have any interest in the field.

This is really not unique to math, by the way. AI is undermining a lot of creative work. Why blog when you have much better odds of making it to the top of HN with autogenerated blog-slop? Why write books when many nonfiction categories on Amazon are now dominated by AI? The list goes on.

There's plenty of people on HN who think it's nothing new, ignoring the huge change in scale. And those who think this is good because there's no inherent value to human creativity if we can get the same content faster and for less. I disagree.

Re: Tao: Open math problems being non-renewably mined by AI

#109
There seems to be a sense wher mathematicians are gamifying math, but are frustrated that AI labs are better at gamifying math.

If an AI solves a problem in an unenlightening way, then there's no reason for mathematicians to stop studying it. Pythagoream Theorem has hundreds of different proofs!

If an AI solves a problem in an enlightening way, mathematicians should study it and propose extensions.

Re: Tao: Open math problems being non-renewably mined by AI

#110
post #64
post #60

Earlier quoted context omitted.

Mathematicians will be less likely to work on a problem if there is a solution - even an incomprehensible one.

> Mathematicians will be less likely to work on a problem if there is a solution Yes, that is Tao's premise, I'm just not sure I buy it. Suppose an oracle existed which could answer any question truthfully. Let's ignore the mechanics of this for now, but it could say things like "the Riemann hypothesis is False" or whatever and we would take it as gospel. Does this mean that we wouldn't have mathematicians or physici…

Yes, the present developments, and the present approach, mean we will not have mathematicians any more.
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